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I'm looking at a theorem in my Analysis textbook that says: If $\mu$ is a complex measure on $X$, then $|\mu|(X) < \infty$.

I can't seem to get my head around this being true. The following seems to me like a counterexample: let $\mu$ be any positive measure with $\mu(X) = \infty$. Then $|\mu|(X) \ge |\mu(X)| = \infty$, so $|\mu|(X) = \infty$.

What am I missing? Can someone give me intuition about why this theorem is true?

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up vote 2 down vote accepted

Read the definition of "complex measure" carefully: typically it requires that the measure of every set be a (finite!) complex number. So by that definition your $\mu$ is not a complex measure.

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